$$$960 e^{\frac{x}{120}}$$$ 的积分

求$$$\int 960 e^{\frac{x}{120}}\, dx$$$。

对 $$$c=960$$$ 和 $$$f{\left(x \right)} = e^{\frac{x}{120}}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$${\color{red}{\int{960 e^{\frac{x}{120}} d x}}} = {\color{red}{\left(960 \int{e^{\frac{x}{120}} d x}\right)}}$$

设$$$u=\frac{x}{120}$$$。

则$$$du=\left(\frac{x}{120}\right)^{\prime }dx = \frac{dx}{120}$$$ (步骤见»)，并有$$$dx = 120 du$$$。

所以，

$$960 {\color{red}{\int{e^{\frac{x}{120}} d x}}} = 960 {\color{red}{\int{120 e^{u} d u}}}$$

对 $$$c=120$$$ 和 $$$f{\left(u \right)} = e^{u}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

$$960 {\color{red}{\int{120 e^{u} d u}}} = 960 {\color{red}{\left(120 \int{e^{u} d u}\right)}}$$

指数函数的积分为 $$$\int{e^{u} d u} = e^{u}$$$：

$$115200 {\color{red}{\int{e^{u} d u}}} = 115200 {\color{red}{e^{u}}}$$

回忆一下 $$$u=\frac{x}{120}$$$:

$$115200 e^{{\color{red}{u}}} = 115200 e^{{\color{red}{\left(\frac{x}{120}\right)}}}$$

因此，

$$\int{960 e^{\frac{x}{120}} d x} = 115200 e^{\frac{x}{120}}$$

加上积分常数：

$$\int{960 e^{\frac{x}{120}} d x} = 115200 e^{\frac{x}{120}}+C$$

$$$\int 960 e^{\frac{x}{120}}\, dx = 115200 e^{\frac{x}{120}} + C$$$A